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Projection methods for large Lyapunov matrix equations. (English) Zbl 1094.65039

Krylov subspace methods are proposed for solving large Lyapunov matrix algebraic equations of the form \(AX + XA^T + BB^T = 0\) where \(A\) and \(B\) are real \(n \times n\) and \(n \times s\) matrices, respectively, with \(s << n\). Equations of this kind appear in many problems of control theory such as computation of the Hankel singular values, model reduction and solution of matrix Riccati equations. The methods proposed are based on the Arnoldi process. It is shown how to extract low rank approximate solutions to Lyapunov equations and expressions are derived for the backward error. Two numerical experiments involving solutions of large Lyapunov equations are presented.
There is no discussion on the connection between the numerical properties of the methods proposed and the conditioning of the Lyapunov equations solved.

MSC:

65F30 Other matrix algorithms (MSC2010)
65F10 Iterative numerical methods for linear systems
15A24 Matrix equations and identities

Software:

Algorithm 432
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References:

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